Rank-one actions, their $(C,F)$-models and constructions with bounded parameters

TL;DR

This paper studies rank-one actions of countable groups, characterizing their properties via $(C,F)$-parameters, and explores their conjugacy, ergodicity, rigidity, and measure-theoretic features, providing new classifications and connections to existing dynamical systems.

Contribution

It introduces a $(C,F)$-model framework for rank-one actions, characterizes conjugacy and ergodic properties, and links these models to classical measure-preserving actions, advancing understanding of group actions on Cantor sets.

Findings

Each topological $(C,F)$-action is a free minimal amenable action with a unique invariant Radon measure.

Necessary and sufficient conditions for conjugacy of $(C,F)$-actions are established.

Conditions for rigidity and total ergodicity of rank-one transformations with bounded parameters are provided.

Abstract

Let $G$ be a discrete countable infinite group. We show that each topological $(C,F)$-action $T$ of $G$ on a locally compact non-compact Cantor set is a free minimal amenable action admitting a unique up to scaling non-zero invariant Radon measure (answer to a question by Kellerhals, Monod and R{\o}rdam). We find necessary and sufficient conditions under which two such actions are topologically conjugate in terms of the underlying $(C,F)$-parameters. If $G$ is linearly ordered Abelian then the topological centralizer of $T$ is trivial. If $G$ is monotileable and amenable, denote by ${\cal A}_G$ the set of all probability preserving actions of $G$ on the unit interval with Lebesgue measure and endow it with the natural topology. We show that the set of $(C,F)$-parameters of all $(C,F)$-actions of $G$ furnished with a suitable topology is a model for ${\cal A}_G$ in the sense of Forman, Rudolph and Weiss. If $T$ is a rank-one transformation with bounded sequences of cuts and spacer maps then we found simple necessary and sufficient conditions on the related $(C,F)$-parameters under which (i) $T$ is rigid, (ii) $T$ is totally ergodic. It is found an alternative proof of Ryzhikov's theorem that if $T$ is totally ergodic and non-rigid rank-one map with bounded parameters then $T$ has MSJ. We also give a more general version of the criterium (by Gao and Hill) for isomorphism and disjointness of two commensurate non-rigid totally ergodic rank-one maps with bounded parameters. It is shown that the rank-one transformations with bounded parameters and no spacers over the last subtowers is a proper subclass of the rank-one transformations with bounded parameters.